Poisson vs Wigner-Dyson level spacing at the chaos onset
The quantum kicked rotor transitions from integrable (K < K_c ≈ 0.97) to chaotic above K_c.
In the integrable regime, eigenvalue spacings follow Poisson statistics P(s) = e^{−s} — levels
can be arbitrarily close (no repulsion). In the chaotic regime, GOE Wigner surmise governs:
P(s) = (π/2)s·exp(−πs²/4) — level repulsion pushes spacings away from zero.
This transition is a fingerprint of quantum chaos (Berry-Tabor conjecture, Bohigas-Giannoni-Schmit).